Solved: 6. Problem 2.6 (from textbook) 2.6 A stenosis is a narrowing of a bloo

Question

6. Problem 2.6 (from textbook) 2.6 A stenosis is a narrowing of a blood vessel or i valve. Stenosis of a blood vessel arises during athero- sclerosis and could occlude an artery, depriving the tissue downstream of oxygen. Further, the fluid shear stresses acting on the endothelial cells lining blood vessels may affect the expression of genes that regulate endothelial cell function. Such gene expression can influence whether the stenosis grows or not. Consider a symmetric stenosis as shown in Figure 2.36. Assume that the velocity profile within the stenosis of radius R(2) has the same shape as the profile outside the stenosis and is represented as ×(1-²) (2.10.5) Outside the stenosis, the radius equals Ro and the max- imum velocity is constant. Within the stenosis, the radius of the fluid channel R(2) equals R(2) Ro - Ro{1 – 0.5[1 − 4(²-)²]¹^²} (2.10.6) v_(r) = V max The origin of the z axis is the midpoint of the stenosis. FIGURE 2.36 Flow through a stenosis. (a) Develop an expression for Vmax in a stenosis in terms of the volumetric flow rate Q, cylindrical tube of radius Ro, and distance along the stenosis z/L. (b) Compute the shear stress acting on the surface of the stenosis (r = R₁) at z = 0 relative to the value outside the stenosis.